The thermally conductive response of the sensor of the present invention to pressure change is believed to depend substantially on the corresponding change of gas density in the vicinity of the thin edge of a hot silicon nitride film where a strong, non-linear temperature gradient exists. It is well known that there is no thermally conductive response to pressure change if the thermal gradient is linear, that is, if the temperature profile has a constant or nearly constant slope and if the molecular mean free path is relatively small. It was not well known, nor was it obvious prior to our experimental realization of the effect in thin films, that the non-linear thermal gradient could play a significant role in pressure sensing. We will now explain the reason for the effect of the non-linear thermal gradient on the pressure response.
FIG. 1b depicts on the left a hot surface, S.sub.h at a temperature T.sub.o, and on the right a cold surface, S.sub.c. All molecules striking the hot surface are assumed to equilibrate with temperature T.sub.o before rebounding. Shown also is a linear temperature profile, G.sub.2, as for planar surfaces, compared with a non-linear profile, G.sub.3 associated with a sharply curved hot surface.
Consider first the linear gradient, and take the molecular mean free path to be a length of X.sub.1. Molecules leaving point X.sub.1 will arrive at the surface with an average temperature of T.sub.1 and will cool the surface in proportion to the difference T.sub.o -T.sub.1. If the pressure is reduced by a factor of two, the mean free path doubles to a length X.sub.2, and molecules that formerly struck the surface from point X.sub.1 now strike it from point X.sub.2, and have a temperature, T.sub.2, such that T.sub.o -T.sub.2 =2(T.sub.o -T.sub.1). Now each molecule reaching the surface has twice the cooling capability, but exact compensation occurs because the molecular density is reduced by a factor of two. Consequently the rate of heat transport is unchanged, and will remain so until the mean free path length approaches the spacing between the two surfaces. Beyond this critical length, the above described compensation fails and the thermal conductance of the gas decreases with decreasing pressure until the density of the gas is so low that no appreciable cooling occurs.
Next consider the curved temperature profile, G.sub.3, having the non-linear gradient as shown. In this case, reducing the pressure by a factor of two and thus doubling the mean free path to a length equal to X.sub.2 does not compensate. It does not proportionally increase the temperature differential because, clearly, (T.sub.o -T.sub.3)&lt;2(T.sub.o -T.sub.1). Therefore the cooling capability of the molecules reaching the hot surface is less in the case of the non-linear gradient as the pressure is reduced, even when the mean free path is very short compared to the spacing between the surfaces. In effect, the thermal gradient near the surface is reduced as the pressure is reduced.
The magnitude of the pressure effect is proportional to the non-linearity that is achieved by shapring the hot surface into a sharp point or edge. However the magnitude of the total effect in all the gas surrounding the hot surface depends on the sum of all conductance paths, the more linear paths diluting the effect of the strongest non-linear regions. Ideally, all the thermal conductance of the gas around the heated bridge microstructure should occur in the strongly non-linear region. Conductance elsewhere in the more linear gradient regions is little changed by gas density variation, and therefore dilutes the corresponding total conductance variation. It is desirable to maximize the non-linear proportion of the total conduction, but before discussing this it is informative to compare the operation of the present invention with the well known Pirani and thermocouple vacuum gauges of the prior art.
The thermocouple and Pirani gauges also use a thermal conductance principle for measuring pressure. They operate typically in the 1-1000 micron (about 10.sup.-6 to 10.sup.-3 atmosphere) range. In these devices the useful operation begins when the gas density falls to the point at which the mean free path of the gas molecules increases to a critical length comparable to the heated wire diameter, and which is analogous to the critical length between planar surfaces. Useful operation ends when the gas density falls further to the point at which the mean free path is substantially larger than the gauge housing. The major effect contributing to the response is the reduction in the density of molecules below the critical density required to compensate and to maintain constant thermal conductance. This occurs when the mean free path length of the molecules exceeds the diameter of the heater wire. A minor effect also contributing to the response is the effect of the non-linear gradient near the heated wire surface. The thermal gradient around a wire varies as 1/r, where r is the distance from the center of the wire. Because the wire is suspended openly without any nearby heat sink, however, the non-linear gradient is small near the wire surface, and its effect on the total conductance is diluted by the long, much more linear remainder of the conductance path to the housing wall over most of which r is large and the gradient is more linear. For example, the wire radius might be 0.002", and the distance from the wire to the housing wall might be 0.500", with the most effective non-linear region within a few thousandths of an inch of the wire. Thus, the effect of the non-linear gradient on the total responses of the piror art gauges is of limited significance.
The sensor of the present invention is distinguished from the above discussed prior art gauges in that it is responsive to pressure change over a range from about 10.sup.-4 atmosphere to 10 atmospheres pressure in contrast to the prior art gauges whose range is within 10.sup.-6 to 10.sup.-3 atmosphere. The present invention is also distinguished from the prior art gauges in that the principal contribution to its pressure response over all but the lowest part of the pressure range is due to an enhanced, non-linear gradient effect. Only when the molecular mean free path becomes one micron or longer below 0.1 atmosphere near the lowest part of its pressure range does the critical length of the molecular mean free path significantly affect the response.
This greatly expanded operating pressure range is due in part to the approximately 100 times smaller thickness of the heated microstructure bridge compared to the prior art wire diameter. Thus our invention provides a much stronger and more non-linear thermal gradient. An equally significant factor is the presence of a nearby heat sink. We have found that the non-linear gradient is further strengthened and the operating pressure range is increased by placing the edge of the heated microstructure bridge very close to a cold heat sink, for example, within a 1-3 micron distance. This configuration enables the conductance through the strong, non-linear region to be a large fraction of the total conductance of the paths near the edge. Because the close edge spacing makes the gradient very large there, a substantial part of the total thermal conduction from the heated microstructure bridge occurs in the edge region, and therefore a substantial fraction of the total conduction also occurs through the strongest non-linear region. Consequently, the change of total conductance corresponding to a pressure change is quite large, and the large magnitude of the non-linear gradient makes the sensor responsive to pressures up to at least ten atmospheres which is about ten thousand times higher than the upper end of the pressure range of the prior art Pirani and thermocouple gauges.
Although the entire heat flow from the heated microstructure bridge should be through the strong non-linear thermal gradient region, in practice it is difficult to design and fabricate a microstructure bridge in which this objective is attained. The ideal geometry would be a microstructure wire of circular cross section within a hollow cylinder heat sink, with the wire radius and spacing from the sink in the micron range, and optimumly selected for the desired pressure range.
We know of no practical method for the fabrication of such a wire and cylinder microstructure. However, microstructure techniques that we have developed do enable the fabrication of heated films of rectangular cross section having thicknesses of less than one micron, and having the film edges spaced one micron or less, or a few microns from a colder heat sink consisting of a silicon edge or corner, or a cold metal film edge, or a cold nitride film edge, or a combination of these. Dielectrics other than nitride could also be used. Thus the non-linear effect can be achieved in the desired range of dimensions, but with a part of the total heat flow also occurring through regions away from the edge where the gradient is more linear.
To maximize the non-linear effect, the edges of the microstructure bridge should be close to the cold heat sink with minimal heat flow through the broad areas of smaller gradients away from the edges. Thus the width of the microstructure bridge should be as small as possible consistent with fabrication tolerances of the design in which the heater film is preferably, but not necessarily, passivated by enclosure within silicon nitride or some other dielectric. Three designs embodying these principles are shown in FIGS. 1, 2 and 3, and will be described in detail.